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Nanoparticle Charge in Fluid from Atomic Force Microscopy Forces within the Nonlinear Poisson-Boltzmann Equation
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作者 Fredy R. Zypman 《Journal of Applied Mathematics and Physics》 2018年第6期1315-1323,共9页
We consider the problem of measuring the electric charge of nanoparticles immersed in a fluid electrolyte. We develop a mathematical framework based on the solution of the nonlinear Poisson-Boltzmann equation to obtai... We consider the problem of measuring the electric charge of nanoparticles immersed in a fluid electrolyte. We develop a mathematical framework based on the solution of the nonlinear Poisson-Boltzmann equation to obtain interaction forces between nanoparticles immersed in a fluid electrolyte and an Atomic Force Microscopy micro spherical probe. This force-separation information is shown explicitly to depend on the charge of the nanoparticle.? This method overcomes the statistical nature of extant methods and renders a charge value for an individual single nanoparticle. 展开更多
关键词 nonlinear Differential equations poisson-boltzmann equation ATOMIC Force Microscopy COLLOIDAL Interaction DEBYE Length
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THE EXACT MEROMORPHIC SOLUTIONS OF SOME NONLINEAR DIFFERENTIAL EQUATIONS
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作者 刘慧芳 毛志强 《Acta Mathematica Scientia》 SCIE CSCD 2024年第1期103-114,共12页
We find the exact forms of meromorphic solutions of the nonlinear differential equations■,n≥3,k≥1,where q,Q are nonzero polynomials,Q■Const.,and p_(1),p_(2),α_(1),α_(2)are nonzero constants withα_(1)≠α_(2).Co... We find the exact forms of meromorphic solutions of the nonlinear differential equations■,n≥3,k≥1,where q,Q are nonzero polynomials,Q■Const.,and p_(1),p_(2),α_(1),α_(2)are nonzero constants withα_(1)≠α_(2).Compared with previous results on the equation p(z)f^(3)+q(z)f"=-sinα(z)with polynomial coefficients,our results show that the coefficient of the term f^((k))perturbed by multiplying an exponential function will affect the structure of its solutions. 展开更多
关键词 Nevanlinna theory nonlinear differential equations meromorphic functions entire functions
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THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS
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作者 黄先勇 邓勋环 王其如 《Acta Mathematica Scientia》 SCIE CSCD 2024年第3期925-946,共22页
In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existe... In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results. 展开更多
关键词 nonlinear delay dynamic equations NONOSCILLATION asymptotic behavior Philostype oscillation criteria generalized Riccati transformation
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Dynamics of fundamental and double-pole breathers and solitons for a nonlinear Schrodinger equation with sextic operator under non-zero boundary conditions
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作者 Luyao Zhang Xiyang Xie 《Chinese Physics B》 SCIE EI CAS CSCD 2024年第9期268-280,共13页
We study the dynamics of fundamental and double-pole breathers and solitons for the focusing and defocusing nonlinear Schrodinger equation with the sextic operator under non-zero boundary conditions. Our analysis main... We study the dynamics of fundamental and double-pole breathers and solitons for the focusing and defocusing nonlinear Schrodinger equation with the sextic operator under non-zero boundary conditions. Our analysis mainly focuses onthe dynamical properties of simple- and double-pole solutions. Firstly, through verification, we find that solutions undernon-zero boundary conditions can be transformed into solutions under zero boundary conditions, whether in simple-pole ordouble-pole cases. For the focusing case, in the investigation of simple-pole solutions, temporal periodic breather and thespatial-temporal periodic breather are obtained by modulating parameters. Additionally, in the case of multi-pole solitons,we analyze parallel-state solitons, bound-state solitons, and intersecting solitons, providing a brief analysis of their interactions.In the double-pole case, we observe that the two solitons undergo two interactions, resulting in a distinctive “triangle”crest. Furthermore, for the defocusing case, we briefly consider two situations of simple-pole solutions, obtaining one andtwo dark solitons. 展开更多
关键词 double-pole solitons double-pole breathers Riemann-Hilbert problem non-zero boundary con-ditions nonlinear Schrodinger equation with sextic operator
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Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation
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作者 Dahe Feng Jibin Li Airen Zhou 《Applied Mathematics》 2024年第8期543-567,共25页
For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits ... For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation. 展开更多
关键词 (2 + 1)-Dimensional nonlinear Dispersive Boussinesq equation BIFURCATIONS Phase Portrait Analytical Periodic Wave Solution Periodic Cusp Wave Solution
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On Two Types of Stability of Solutions to a Pair of Damped Coupled Nonlinear Evolution Equations
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作者 Mark Jones 《Advances in Pure Mathematics》 2024年第5期354-366,共13页
The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid... The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid dynamics, fibre optics or electron plasmas. The main result is that any small perturbation to the solution remains small for all time. Here small is interpreted as being both in the supremum sense and the square integrable sense. 展开更多
关键词 nonlinear Schrödinger equation STABILITY Capillary-Gravity Waves
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A Comparative Study of Adomian Decomposition Method with Variational Iteration Method for Solving Linear and Nonlinear Differential Equations
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作者 Sarah Khaled Al Baghdadi N. Ameer Ahammad 《Journal of Applied Mathematics and Physics》 2024年第8期2789-2819,共31页
This study compares the Adomian Decomposition Method (ADM) and the Variational Iteration Method (VIM) for solving nonlinear differential equations in engineering. Differential equations are essential for modeling dyna... This study compares the Adomian Decomposition Method (ADM) and the Variational Iteration Method (VIM) for solving nonlinear differential equations in engineering. Differential equations are essential for modeling dynamic systems in various disciplines, including biological processes, heat transfer, and control systems. This study addresses first, second, and third-order nonlinear differential equations using Mathematica for data generation and graphing. The ADM, developed by George Adomian, uses Adomian polynomials to handle nonlinear terms, which can be computationally intensive. In contrast, VIM, developed by He, directly iterates the correction functional, providing a more straightforward and efficient approach. This study highlights VIM’s rapid convergence and effectiveness of VIM, particularly for nonlinear problems, where it simplifies calculations and offers direct solutions without polynomial derivation. The results demonstrate VIM’s superior efficiency and rapid convergence of VIM compared with ADM. The VIM’s minimal computational requirements make it practical for real-time applications and complex system modeling. Our findings align with those of previous research, confirming VIM’s efficiency of VIM in various engineering applications. This study emphasizes the importance of selecting appropriate methods based on specific problem requirements. While ADM is valuable for certain nonlinearities, VIM’s approach is ideal for many engineering scenarios. Future research should explore broader applications and hybrid methods to enhance the solution’s accuracy and efficiency. This comprehensive comparison provides valuable guidance for selecting effective numerical methods for differential equations in engineering. 展开更多
关键词 Differential equations Numerical Analysis Mathematical Computing Engineering Models nonlinear Dynamics
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Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations
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作者 Dechao Gao Zeshan Qiu +1 位作者 Lizan Wang Jianxin Li 《Journal of Applied Mathematics and Physics》 2024年第4期1089-1100,共12页
The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference oper... The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L<sub>2</sub>-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective. 展开更多
关键词 Crank-Nicolson Quasi-Compact Scheme Fractional Advection-Diffusion equations nonlinear Stability and Convergence
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Existence and Stability of Standing Waves for the Nonlinear Schrödinger Equation with Combined Nonlinearities and a Partial Harmonic Potential
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作者 Wei Wang 《Journal of Applied Mathematics and Physics》 2024年第5期1606-1615,共10页
In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercriti... In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential. 展开更多
关键词 nonlinear Schrödinger equation Orbital Stability Standing Waves
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Stability of Standing Waves for the Nonlinear Schrödinger Equation with Mixed Power-Type and Hartree-Type Nonlinearities
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作者 Chunyang Yan 《Journal of Applied Mathematics and Physics》 2024年第10期3439-3457,共19页
This paper studies the existence of stable standing waves for the nonlinear Schrödinger equation with Hartree-type nonlinearity i∂tψ+Δψ+| ψ |pψ+(| x |−γ∗| ψ |2)ψ=0,   (t,x)∈[ 0,T )×ℝN.Where ψ=ψ(t,... This paper studies the existence of stable standing waves for the nonlinear Schrödinger equation with Hartree-type nonlinearity i∂tψ+Δψ+| ψ |pψ+(| x |−γ∗| ψ |2)ψ=0,   (t,x)∈[ 0,T )×ℝN.Where ψ=ψ(t,x)is a complex valued function of (t,x)∈ℝ+×ℝN. The parameters N≥3, 0p4Nand 0γmin{ 4,N }. By using the variational methods and concentration compactness principle, we prove the orbital stability of standing waves. 展开更多
关键词 nonlinear Schrödinger equation Concentration Compactness Principle Orbital Stability
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Thermomechanical Dynamics (TMD) and Bifurcation-Integration Solutions in Nonlinear Differential Equations with Time-Dependent Coefficients
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作者 Hiroshi Uechi Lisa Uechi Schun T. Uechi 《Journal of Applied Mathematics and Physics》 2024年第5期1733-1743,共11页
The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple ba... The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple back-and-forth motion develops into water drinking motion. The solution to a drinking bird equation of motion manifests itself the transition from thermodynamic equilibrium to nonequilibrium irreversible states. The independent solution signifying a nonequilibrium thermal state seems to be constructed as if two independent bifurcation solutions are synthesized, and so, the solution is tentatively termed as the bifurcation-integration solution. The bifurcation-integration solution expresses the transition from mechanical and thermodynamic equilibrium to a nonequilibrium irreversible state, which is explicitly shown by the nonlinear differential equation with time-dependent coefficients (NDE-TC). The analysis established a new theoretical approach to nonequilibrium irreversible states, thermomechanical dynamics (TMD). The TMD method enables one to obtain thermodynamically consistent and time-dependent progresses of thermodynamic quantities, by employing the bifurcation-integration solutions of NDE-TC. We hope that the basic properties of bifurcation-integration solutions will be studied and investigated further in mathematics, physics, chemistry and nonlinear sciences in general. 展开更多
关键词 The nonlinear Differential equation with Time-Dependent Coefficients The Bifurcation-Integration Solution Nonequilibrium Irreversible States Thermomechanical Dynamics (TMD)
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A FINITE DIFFERENCE SCHEME FOR SOLVING THE NONLINEAR POISSON-BOLTZMANN EQUATION MODELING CHARGED SPHERES 被引量:3
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作者 Zhong-hua Qiao Zhi-lin Li Tao Tang 《Journal of Computational Mathematics》 SCIE CSCD 2006年第3期252-264,共13页
In this work, we propose an efficient numerical method for computing the electrostatic interaction between two like-charged spherical particles which is governed by the nonlinear Poisson-Boltzmann equation. The nonlin... In this work, we propose an efficient numerical method for computing the electrostatic interaction between two like-charged spherical particles which is governed by the nonlinear Poisson-Boltzmann equation. The nonlinear problem is solved by a monotone iterative method which leads to a sequence of linearized equations. A modified central finite difference scheme is developed to solve the linearized equations on an exterior irregular domain using a uniform Cartesian grid. With uniform grids, the method is simple, and as a consequence, multigrid solvers can be employed to speed up the convergence. Numerical experiments on cases with two isolated spheres and two spheres confined in a charged cylindrical pore are carried out using the proposed method. Our numerical schemes are found efficient and the numerical results are found in good agreement with the previous published results. 展开更多
关键词 nonlinear poisson-boltzmann equation Electrostatic interaction Irregulardomain Monotone iterative method Multigrid solver
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GLOBAL SOLUTIONS IN THE CRITICAL SOBOLEV SPACE FOR THE LANDAU EQUATION
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作者 Hao WANG 《Acta Mathematica Scientia》 SCIE CSCD 2024年第4期1347-1372,共26页
The Landau equation is studied for hard potential with-2≤γ≤1.Under a perturbation setting,a unique global solution of the Cauchy problem to the Landau equation is established in a critical Sobolev space H_(x)^(d)L_... The Landau equation is studied for hard potential with-2≤γ≤1.Under a perturbation setting,a unique global solution of the Cauchy problem to the Landau equation is established in a critical Sobolev space H_(x)^(d)L_(v)^(2)(d>3/2),which extends the results of[11]in the torus domain to the whole space R_(x)^(3).Here we utilize the pseudo-differential calculus to derive our desired result. 展开更多
关键词 Landau equation nonlinear energy method global existence pseudo-differential calculus
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NORMALIZED SOLUTIONS FOR THE GENERAL KIRCHHOFF TYPE EQUATIONS
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作者 Wenmin LIU Xuexiu ZHONG Jinfang ZHOU 《Acta Mathematica Scientia》 SCIE CSCD 2024年第5期1886-1902,共17页
In the present paper,we prove the existence,non-existence and multiplicity of positive normalized solutions(λ_(c),u_(c))∈R×H^(1)(R^(N))to the general Kirchhoff problem-M■,satisfying the normalization constrain... In the present paper,we prove the existence,non-existence and multiplicity of positive normalized solutions(λ_(c),u_(c))∈R×H^(1)(R^(N))to the general Kirchhoff problem-M■,satisfying the normalization constraint f_(R)^N u^2dx=c,where M∈C([0,∞))is a given function satisfying some suitable assumptions.Our argument is not by the classical variational method,but by a global branch approach developed by Jeanjean et al.[J Math Pures Appl,2024,183:44–75]and a direct correspondence,so we can handle in a unified way the nonlinearities g(s),which are either mass subcritical,mass critical or mass supercritical. 展开更多
关键词 normalized solution Kirchhoff type equations general nonlinearities asymptotic behavior
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Analysis of Extended Fisher-Kolmogorov Equation in 2D Utilizing the Generalized Finite Difference Method with Supplementary Nodes
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作者 Bingrui Ju Wenxiang Sun +1 位作者 Wenzhen Qu Yan Gu 《Computer Modeling in Engineering & Sciences》 SCIE EI 2024年第10期267-280,共14页
In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolso... In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme.Following temporal discretization,the generalized finite difference method(GFDM)with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node.These supplementary nodes are distributed along the boundary to match the number of boundary nodes.By incorporating supplementary nodes,the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation.To demonstrate the efficacy of our approach,we present three numerical examples showcasing its performance in solving this nonlinear problem. 展开更多
关键词 Generalized finite difference method nonlinear extended Fisher-Kolmogorov equation Crank-Nicolson scheme
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MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHR?DINGER EQUATION IN A HIGH DIMENSION
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作者 钟延生 李永青 《Acta Mathematica Scientia》 SCIE CSCD 2023年第4期1819-1840,共22页
The existence and multiplicity of positive solutions for equation(1.1)with the new critical exponent 4<p<2·2*shall be investigated in a high dimension.The conclusions extend the relative results recently at... The existence and multiplicity of positive solutions for equation(1.1)with the new critical exponent 4<p<2·2*shall be investigated in a high dimension.The conclusions extend the relative results recently attained in[1]for the one-dimensional case.More precisely,as the coefficient a(x)in the nonlinearity is sign-changing,the modified term 2(Δ(|u|2))u is still helpful for obtaining multiple positive solutions in a high dimension,even if a sign condition like∫_(R)N a(x)e_(1)^(p)dx<0(also named“a necessary condition”see[2,3])does not hold. 展开更多
关键词 variational methods nonlinear Schrodinger equation sign-changing
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Coupled-generalized nonlinear Schr¨odinger equations solved by adaptive step-size methods in interaction picture
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作者 陈磊 李磐 +3 位作者 刘河山 余锦 柯常军 罗子人 《Chinese Physics B》 SCIE EI CAS CSCD 2023年第2期332-340,共9页
We extend two adaptive step-size methods for solving two-dimensional or multi-dimensional generalized nonlinear Schr ¨odinger equation(GNLSE): one is the conservation quantity error adaptive step-control method(R... We extend two adaptive step-size methods for solving two-dimensional or multi-dimensional generalized nonlinear Schr ¨odinger equation(GNLSE): one is the conservation quantity error adaptive step-control method(RK4IP-CQE), and the other is the local error adaptive step-control method(RK4IP-LEM). The methods are developed in the vector form of fourthorder Runge–Kutta iterative scheme in the interaction picture by converting a vector equation in frequency domain. By simulating the supercontinuum generated from the high birefringence photonic crystal fiber, the calculation accuracies and the efficiencies of the two adaptive step-size methods are discussed. The simulation results show that the two methods have the same global average error, while RK4IP-LEM spends more time than RK4IP-CQE. The decrease of huge calculation time is due to the differences in the convergences of the relative photon number error and the approximated local error between these two adaptive step-size algorithms. 展开更多
关键词 nonlinear optics optical propagation in nonlinear media coupled-generalized nonlinear Schr?dinger equations(C-GNLSE) adaptive step-size methods
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Quantitative analysis of soliton interactions based on the exact solutions of the nonlinear Schr?dinger equation
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作者 张雪峰 许韬 +1 位作者 李敏 孟悦 《Chinese Physics B》 SCIE EI CAS CSCD 2023年第1期244-252,共9页
We make a quantitative study on the soliton interactions in the nonlinear Schro¨dinger equation(NLSE) and its variable–coefficient(vc) counterpart. For the regular two-soliton and double-pole solutions of the NL... We make a quantitative study on the soliton interactions in the nonlinear Schro¨dinger equation(NLSE) and its variable–coefficient(vc) counterpart. For the regular two-soliton and double-pole solutions of the NLSE, we employ the asymptotic analysis method to obtain the expressions of asymptotic solitons, and analyze the interaction properties based on the soliton physical quantities(especially the soliton accelerations and interaction forces);whereas for the bounded two-soliton solution, we numerically calculate the soliton center positions and accelerations, and discuss the soliton interaction scenarios in three typical bounded cases. Via some variable transformations, we also obtain the inhomogeneous regular two-soliton and double-pole solutions for the vcNLSE with an integrable condition. Based on the expressions of asymptotic solitons, we quantitatively study the two-soliton interactions with some inhomogeneous dispersion profiles,particularly discuss the influence of the variable dispersion function f(t) on the soliton interaction dynamics. 展开更多
关键词 nonlinear Schr?dinger equation soliton solutions asymptotic analysis soliton interactions
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Breather and its interaction with rogue wave of the coupled modified nonlinear Schrodinger equation
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作者 王明 徐涛 +1 位作者 何国亮 田雨 《Chinese Physics B》 SCIE EI CAS CSCD 2023年第5期350-356,共7页
We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localiz... We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localized wave solution,the N-fold generalized Darboux transformation is given.Under the condition that the characteristic equation admits a double-root,we present the expression of the first-order interactional solution.Then we graphically analyze the dynamics of the breather and rogue wave.Due to the simultaneous existence of nonlinear and self-steepening terms in the equation,different profiles in two components for the breathers are presented. 展开更多
关键词 coupled modified nonlinear Schr?dinger equation Darboux transformation BREATHER rouge wave
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Nonlinear Algebraic Equations Solved by an Optimal Splitting-Linearizing Iterative Method
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作者 Chein-Shan Liu Essam REl-Zahar Yung-Wei Chen 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第5期1111-1130,共20页
How to accelerate the convergence speed and avoid computing the inversion of a Jacobian matrix is important in the solution of nonlinear algebraic equations(NAEs).This paper develops an approach with a splitting-linea... How to accelerate the convergence speed and avoid computing the inversion of a Jacobian matrix is important in the solution of nonlinear algebraic equations(NAEs).This paper develops an approach with a splitting-linearizing technique based on the nonlinear term to reduce the effect of the nonlinear terms.We decompose the nonlinear terms in the NAEs through a splitting parameter and then linearize the NAEs around the values at the previous step to a linear system.Through the maximal orthogonal projection concept,to minimize a merit function within a selected interval of splitting parameters,the optimal parameters can be quickly determined.In each step,a linear system is solved by the Gaussian elimination method,and the whole iteration procedure is convergent very fast.Several numerical tests show the high performance of the optimal split-linearization iterative method(OSLIM). 展开更多
关键词 nonlinear algebraic equations novel splitting-linearizing technique iterative method maximal projection optimal splitting parameter
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